If \(\mathbf {P}\)  is regular and \(\mathrm {dim}(\mathcal {M}(\mathbf {y},A,\mathbf {P})) \le 1\) , then \(\mathcal {M}_{\lbrace \Gamma _i\rbrace }(\mathbf {y},A,\mathsf {t},\mathbf {P}) = \emptyset \)  unless \(\mathsf {t}=\mathsf {t}^{\mathsf {can}}\)  and \(I=\emptyset \) .


Observe that the subset \(\mathcal {M}^*_{\lbrace \Gamma _i\rbrace }(\mathbf {y},A,\mathsf {t},\mathbf {P})\)  of discs with simple bubble trees attached is empty by Lemma REF , since it is regular and has negative virtual dimension.
On the other hand, one can always replace a non-simple bubble tree with a simple one (see [1]}). The procedure involves replacing multiply-covered spheres by their image, deleting some spheres, and rearranging the marked points; in particular, it cannot increase the Chern number by Corollary REF . It follows that the moduli spaces of discs with non-simple bubble trees attached are also empty.
